# Subspace related definition.

Definition: Let $$S$$ be a set of vectors in a vector space $$V$$. The Subspace Spanned by $$S$$ is defined to be the intersection $$W$$ of all subspace of $$V$$ which contain $$S$$. When $$S$$ is a finite set of vectors, $$S$$={$$v_1$$,$$v_2$$...$$v_n$$}, we shall simply call $$W$$ the subspace spanned by the vectors $$v_1$$,$$v_2$$...$$v_n$$.

[Linear Algebra,Hoffman,Kunze]

I understand from this definition if $$w_1$$,$$w_2$$...$$w_n$$ be the subspace spanned by $$S$$.

let W=$$w_1\capw_2\cap$$...$$\capw_n$$. then $$W$$ is the smallest subspace spanned by $$S$$ and contain $$S$$.

Is this correct?

Not quite.

Firstly, you assume that there might be finitely many subspaces $$w_1,\cdots,w_n$$ which contain $$S$$. This need not always be true. Consider, for instance, the space $$P := \{ a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n \mid n \in \mathbb{N}, a_i \in \mathbb{R} \}$$ i.e. the space of all polynomials, interpreted over $$\mathbb{R}$$ for simplicity. Then the zero vector certainly forms a trivial, singleton subspace, and it is contained in the collections $$W_n := \text{span}\{x^n\}$$, of which there are infinitely many.

$$w_1$$,$$w_2$$...$$w_n$$ be the subspace spanned by $$S$$.

Moreover, are you claiming this list is a subspace, or collection of subspaces? It is most definitely not a subspace in itself (since they must be closed under linear combination, and most fields have infinitely many elements).

let W=$$w_1\capw_2\cap$$...$$\capw_n$$. then $$W$$ is the [...] subspace spanned by $$S$$ and contain $$S$$.

If, for some reason, we only have finitely many subspaces $$w_1,\cdots,w_n$$ that contain $$S$$, then yes, $$W := \bigcap_{i=1}^n w_i$$ is the subspace spanned by $$S$$.

But if you have infinitely many, or in general $$\{W_i\}_{i \in I}$$ for some arbitrary indexing set $$I$$, which consists of every subspace containing $$S$$, then yes: $$W := \bigcap_{i \in I} W_i$$ is the subspace spanned by $$S$$. More succinctly, if $$W \le V$$ means $$W$$ is a vector subspace of $$V$$, $$\text{span}(S) := \bigcap_{\substack{W \le V \\ \text{such that} \\ S \subseteq V}} W$$

smallest subspace spanned by $$S$$

There is no "largest" or "smallest" subspace; the subspace spanned by $$S$$ is unique, there is only one.

You may recall from a more basic linear algebra class that $$\text{span}\Big( \{v_1,\cdots,v_n\} \Big) = \left\{ \sum_{i=1}^n c_i v_i \, \middle| \, \text{across all possible choices of } c_i \in \mathbb{R} \right\}$$ assuming we're working in a real vector space. It may help of you to think this strange definition of the spanning set as this, but generalized in a way that works for infinite sets and infinite-dimensional vector spaces.

• @PrincessEev Thank you. I claim this is a collection of subspaces. Commented Aug 18, 2023 at 16:19